Precursor (physics)

Precursors are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation (microwaves,visible light, and terahertz radiation) as well as in fluid surface waves and seismic waves.

Precursors were first theoretically predicted in 1914 by Arnold Sommerfeld for the case of electromagnetic radiation propagating through a neutral dielectric in a region of normal dispersion. Sommerfeld's work was expanded in the following years by Léon Brillouin, who applied the saddle point approximation to compute the integrals involved. However, it was not until 1969 that precursors were first experimentally confirmed for the case of microwaves propagating in a waveguide, and much of the experimental work observing precursors in other types of waves has only been done since the year 2000. This experimental lag is mainly due to the fact that in many situations, precursors have a much smaller amplitude than the signals that give rise to them (a baseline figure given by Brillouin is six orders of magnitude smaller). As a result, experimental confirmations could only be done after technology became available to detect precursors.

As a dispersive phenomenon, the amplitude at any distance and time of a precursor wave propagating in one dimension can be expressed by the Fourier integral

where ${\displaystyle {\hat {\zeta }}_{0}(\omega )}$ is the Fourier transform of the initial impulse and the complex exponential ${\displaystyle \exp \left[-i\left(k(\omega )x-\omega t\right)\right]}$ represents the individual component wavelets summed in the integral. To account for the effects of dispersion, the phase of the exponential must include the dispersion relation (here, the ${\displaystyle k(\omega )}$ factor) for the particular medium in which the wave is propagating.

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